20e_04s_2s

# 20e_04s_2s - Math 20E Second Midterm Solutions 1 There may...

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Unformatted text preview: Math 20E Second Midterm Solutions May 26, 2004 1. There may be other answers than these (a) gradient, scalar OR gradient, potential OR curl, solenoidal (b) 0 (c) harmonic (d) gradient, curl OR irrotational function, solenoidal function OR gradient of a scalar potential, curl of a vector potential 2. First solution: Use the divergence theorem and ∇ · ∇ × F = 0. Second solution: Use Stokes’ Theorem. Since we are integrating over a closed surface, there is no boundary and so the integral is zero. 3. Since F is defined everywhere, we can take the domain to be all of 3-space and R = . (a) Since F is homogenous of degree 2, the supplementary homework tells us that G = 12 2 + 2 (2 xz i − z 2 k ) × R = 3( yz 2 i − 3 xz 2 j + 2 xyz k ) . Another way to solve it is to compute the integral in the text: G = integraldisplay 1 t (24 xzt 2 i − 12 z 2 t 2 k ) × R dt = integraldisplay 1 t (12 yz 2 t 2 i − 36 xz 2 t 2 j + 24 xyzt 2 k ) dt = 3 yz 2 i − 9 xz 2 j + 6 xyz k ....
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